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2 ESSENTIAL MAPPINGS IN THE UNIT CYCLE 253

(Ap.2.5) Let E be a compact metric space, I = [0, !],/# continuous mapping
of E x I into U. If the mapping x -+f(x, 0) is essential (resp. inessential),
so is the mapping x ->f(x, 1).

As/is uniformly continuous in E x I (3.16.5), there is an integer n > 1
such that the relation \s - t \ < l/n implies \f(x, s) ~/(x, t)\ < 1 for any x e E.
Let f_k(x) =/(*, k/ri) for 0 ^ k < w; we therefore have \f_k(x) ~f_k+1(x)\ ^ 1
for any x e E and 0 < k < n — 1, and as \f_k(x)\ = IA+i(*)l = 1 for x e E,
we have/_fc(.x) ^ —A+i(*) for jc e E. Hence the result by (Ap.2.4).

(Ap.2.6) Any continuous mapping f of a closed ball (in R") into U is
inessential.

Let E be the ball d(x₉ a) ^ r, and define #(x, 0 —f(a + t(x — a)); then
g is continuous in E x [0, 1], g(x, 1) =f(x) and #(x, 0) —f(a); as ^c-^^(jc, 0)
is inessential (Ap.2.3), so is/by (Ap.2.5).

(Ap.2.7) Let A, B ie two closed subsets of a metric space E, such that
E = A u B, and that A n B be connected. Let f be a continuous mapping of
E into U; if the restrictions of f to A and B are inessential, f is inessential

There are, by assumption, continuous mappings g, h of A and B into R
such that f(x) = e^ig(x} in A, f(x) = e^f/l(x) in B. For x e A n B, we have
therefore e^id(x) = e^lh(x\ hence (9.5.5) (g(x) - h(x))/2n is an integer; but
as g — h is continuous in the connected set A n B, this implies g — h is a
constant 2nn in A n B by (3.19.7). Let then u(x) = g(x) in A, u(x) = h(x) + 2/77T
in B; it is clear that f(x) = <?^IM(X) in E, and as g and h + 2mt coincide in
A n B, u is continuous in E. Q.E.D.

(Ap.2.8) In order that a continuous mapping f of "U into itself be essential,
a necessary and sufficient condition is that y(0; y) 9* Qfor the loop y: t -»/(e^If)
(0 < / < 2n).

By (9.8.1) we can write /(c^tt) = e'^⁽⁰, where \l/ is continuous in [0, 2?i],
and \l/(2n) - \I/(Q) = 2nn by (9.5.5), n being the index J(0;y). Let
fl)(r,f) = ^(f) + {(^ + i/r(0)-^(0); if, for C = ^'' (0 < f < 2?r) and for
0 < f < 1, we write #(C, <J) = e^fco(t'^, ^ is continuous in (U - {!}) x [0, 1]
by (9.5.7), and as e''^w<²"'«> = _eMo,« ₌₌y(₁) _for _any ^ ^ _is extended by